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Asymptotic iteration method for eigenvalue problems. (English) Zbl 1070.34113

The main task of the present work is to introduce a new technique to solve second-order homogeneous linear differential equations

y '' =λ 0 (x)y ' +s 0 (x)y,(*)

where λ 0 ,s 0 C (a,b). The following result is established:

The differential equation (*) has the general solution

y(x)=exp- x αdtC 2 +C 1 x exp t (λ 0 (τ)+2α(τ))dτdt

if for some n>0

s n λ n =s n-1 λ n-1 α,

where λ k =λ k-1 ' +s k-1 +λ 0 λ k-1 and s k =s k-1 ' +s 0 λ k-1 for k=1,2,···,n.

Applications to Schrödinger-type problems, including someones with highly singular potentials, are presented.


MSC:
34L16Numerical approximation of eigenvalues and of other parts of the spectrum
34A30Linear ODE and systems, general
81Q10Selfadjoint operator theory in quantum theory, including spectral analysis